The least squares problem of the matrix equation A 1 X 1 B 1 T + A 2 X 2 B 2 T = T

2009 ◽  
Vol 24 (4) ◽  
pp. 451-461 ◽  
Author(s):  
Yu-yang Qiu ◽  
Zhen-yue Zhang ◽  
An-ding Wang
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Least Squares Problem ◽  
The Matrix

Gravity interpretation with the aid of quadratic programming

Geophysics ◽  
1980 ◽  
Vol 45 (3) ◽  
pp. 403-419 ◽  
Author(s):  
N. J. Fisher ◽  
L. E. Howard
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Theoretical Data ◽  
Linear Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Least Squares Problem ◽  
The Matrix ◽  
Gravity Interpretation ◽  
Value Decomposition

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

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